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A remark on central sequence algebras of the tensor product of $ \mathrm{II}_{1}$ factors


Authors: Wenming Wu and Wei Yuan
Journal: Proc. Amer. Math. Soc. 142 (2014), 2829-2835
MSC (2010): Primary 46L10
DOI: https://doi.org/10.1090/S0002-9939-2014-12046-6
Published electronically: May 12, 2014
MathSciNet review: 3209336
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Abstract: Let $ \mathcal {M}$ and $ \mathcal {N}$ be two type $ \mathrm {II}_{1}$ factors with separable predual and $ \omega $ a free ultrafilter on $ \mathbb{N}$. If the central sequence algebra $ \mathcal {N}_{\omega }$ is abelian and there is a non-atomic abelian subalgebra $ \mathcal {A}$ in $ \mathcal {M}$ such that any central sequence of $ \mathcal {M}\overline {\otimes }\mathcal {N}$ is contained in the ultrapower $ (\mathcal {A}\overline {\otimes }\mathcal {N})^{\omega }$, then $ (\mathcal {M}\overline {\otimes }\mathcal {N})_{\omega }$ is abelian. It is also shown that there is an action $ \alpha $ of the free group $ F_2$ on the group von Neumann algebra $ \mathcal {L}_{\mathbb{Z}}$ such that the central sequence algebra of $ \mathcal {M}=\mathcal {L}_{\mathbb{Z}}\rtimes _{\alpha } F_2$ is abelian and non-trivial and any central sequence in $ \mathcal {M}\overline {\otimes }\mathcal {N}$ is in the ultrapower $ (\mathcal {L}_{\mathbb{Z}}\overline {\otimes }\mathcal {N})^{\omega }$.


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Additional Information

Wenming Wu
Affiliation: College of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, People’s Republic of China
Email: wuwm@amss.ac.cn

Wei Yuan
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, 100084, People’s Republic of China
Email: wyuan@math.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-12046-6
Received by editor(s): November 18, 2011
Received by editor(s) in revised form: September 10, 2012
Published electronically: May 12, 2014
Additional Notes: This work was partially supported by NSFC (No.11271390, No. 11301511) and Natural Science Foundation Project of CQ CSTC (No. CSTC, 2010BB9318).
Communicated by: Marius Junge
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.